Engineering Statistics Handbook Episode 6 Part 11 ppsx

5. Process Improvement 5.2. Assumptions 5.2.1.Is the measurement system capable? Metrology capabilities are a key factor in most experiments It is unhelpful to find, after you have finished all the experimental runs, that the measurement devices you have at your disposal cannot measure the changes you were hoping to see. Plan to check this out before embarking on the experiment itself. Measurement process characterization is covered in Chapter 2. SPC check of measurement devices In addition, it is advisable, especially if the experimental material is planned to arrive for measurement over a protracted period, that an SPC (i.e., quality control) check is kept on all measurement devices from the start to the conclusion of the whole experimental project. Strange experimental outcomes can often be traced to `hiccups' in the metrology system. 5.2.1. Is the measurement system capable? http://www.itl.nist.gov/div898/handbook/pri/section2/pri21.htm [5/1/2006 10:30:19 AM] 5. Process Improvement 5.2. Assumptions 5.2.3.Is there a simple model? Polynomial approximation models only work for smoothly varying outputs In this chapter we restrict ourselves to the case for which the response variable(s) are continuous outputs denoted as Y. Over the experimental range, the outputs must not only be continuous, but also reasonably smooth. A sharp falloff in Y values is likely to be missed by the approximating polynomials that we use because these polynomials assume a smoothly curving underlying response surface. Piecewise smoothness requires separate experiments If the surface under investigation is known to be only piecewise smooth, then the experiments will have to be broken up into separate experiments, each investigating the shape of the separate sections. A surface that is known to be very jagged (i.e., non-smooth) will not be successfully approximated by a smooth polynomial. Examples of piecewise smooth and jagged responses Piecewise Smooth Jagged FIGURE 2.1 Examples of Piecewise Smooth and Jagged Responses 5.2.3. Is there a simple model? http://www.itl.nist.gov/div898/handbook/pri/section2/pri23.htm [5/1/2006 10:30:20 AM] Histogram The histogram is a frequency plot obtained by placing the data in regularly spaced cells and plotting each cell frequency versus the center of the cell. Figure 2.2 illustrates an approximately normal distribution of residuals produced by a model for a calibration process. We have superimposed a normal density function on the histogram. Small sample sizes Sample sizes of residuals are generally small (<50) because experiments have limited treatment combinations, so a histogram is not be the best choice for judging the distribution of residuals. A more sensitive graph is the normal probability plot. Normal probability plot The steps in forming a normal probability plot are: Sort the residuals into ascending order. ● Calculate the cumulative probability of each residual using the formula: P(i-th residual) = i/(N+1) with P denoting the cumulative probability of a point, i is the order of the value in the list and N is the number of entries in the list. ● Plot the calculated p-values versus the residual value on normal probability paper.● The normal probability plot should produce an approximately straight line if the points come from a normal distribution. 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898/handbook/pri/section2/pri24.htm (2 of 10) [5/1/2006 10:30:21 AM] Sample normal probability plot with overlaid dot plot Figure 2.3 below illustrates the normal probability graph created from the same group of residuals used for Figure 2.2. This graph includes the addition of a dot plot. The dot plot is the collection of points along the left y-axis. These are the values of the residuals. The purpose of the dot plot is to provide an indication the distribution of the residuals. "S" shaped curves indicate bimodal distribution Small departures from the straight line in the normal probability plot are common, but a clearly "S" shaped curve on this graph suggests a bimodal distribution of residuals. Breaks near the middle of this graph are also indications of abnormalities in the residual distribution. NOTE: Studentized residuals are residuals converted to a scale approximately representing the standard deviation of an individual residual from the center of the residual distribution. The technique used to convert residuals to this form produces a Student's t distribution of values. Independence of Residuals Over Time Run sequence plot If the order of the observations in a data table represents the order of execution of each treatment combination, then a plot of the residuals of those observations versus the case order or time order of the observations will test for any time dependency. These are referred to as run sequence plots. 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898/handbook/pri/section2/pri24.htm (3 of 10) [5/1/2006 10:30:21 AM] Sample run sequence plot that exhibits a time trend Sample run sequence plot that does not exhibit a time trend 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898/handbook/pri/section2/pri24.htm (4 of 10) [5/1/2006 10:30:21 AM] Interpretation of the sample run sequence plots The residuals in Figure 2.4 suggest a time trend, while those in Figure 2.5 do not. Figure 2.4 suggests that the system was drifting slowly to lower values as the investigation continued. In extreme cases a drift of the equipment will produce models with very poor ability to account for the variability in the data (low R 2 ). If the investigation includes centerpoints, then plotting them in time order may produce a more clear indication of a time trend if one exists. Plotting the raw responses in time sequence can also sometimes detect trend changes in a process that residual plots might not detect. Plot of Residuals Versus Corresponding Predicted Values Check for increasing residuals as size of fitted value increases Plotting residuals versus the value of a fitted response should produce a distribution of points scattered randomly about 0, regardless of the size of the fitted value. Quite commonly, however, residual values may increase as the size of the fitted value increases. When this happens, the residual cloud becomes "funnel shaped" with the larger end toward larger fitted values; that is, the residuals have larger and larger scatter as the value of the response increases. Plotting the absolute values of the residuals instead of the signed values will produce a "wedge-shaped" distribution; a smoothing function is added to each graph which helps to show the trend. 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898/handbook/pri/section2/pri24.htm (5 of 10) [5/1/2006 10:30:21 AM] Sample residuals versus fitted values plot showing increasing residuals Sample residuals versus fitted values plot that does not show increasing residuals 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898/handbook/pri/section2/pri24.htm (6 of 10) [5/1/2006 10:30:22 AM] Interpretation of the residuals versus fitted values plots A residual distribution such as that in Figure 2.6 showing a trend to higher absolute residuals as the value of the response increases suggests that one should transform the response, perhaps by modeling its logarithm or square root, etc., (contractive transformations). Transforming a response in this fashion often simplifies its relationship with a predictor variable and leads to simpler models. Later sections discuss transformation in more detail. Figure 2.7 plots the residuals after a transformation on the response variable was used to reduce the scatter. Notice the difference in scales on the vertical axes. Independence of Residuals from Factor Settings Sample residuals versus factor setting plot 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898/handbook/pri/section2/pri24.htm (7 of 10) [5/1/2006 10:30:22 AM] Sample residuals versus factor setting plot after adding a quadratic term 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898/handbook/pri/section2/pri24.htm (8 of 10) [5/1/2006 10:30:22 AM] [...]... Note: Residuals are an important subject discussed repeatedly in this Handbook For example, graphical residual plots using Dataplot are discussed in Chapter 1 and the general examination of residuals as a part of model building is discussed in Chapter 4 http://www.itl.nist.gov/div898 /handbook/ pri/section2/pri24.htm (10 of 10) [5/1/20 06 10:30:22 AM] 5.3 Choosing an experimental design 2 Box-Behnken designs... is a trace resembling a "frown" or a "smile" in these graphs Sample residuals versus factor setting plot lacking one or more higher-order terms http://www.itl.nist.gov/div898 /handbook/ pri/section2/pri24.htm (9 of 10) [5/1/20 06 10:30:22 AM] 5.2.4 Are the model residuals well-behaved? Interpretation of plot The example given in Figures 2.8 and 2.9 obviously involves five levels of the predictor The experiment... foldover designs 2 Alternative foldover designs 9 Three-level full factorial designs 10 Three-level, mixed level and fractional factorial designs http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3.htm (2 of 2) [5/1/20 06 10:30:22 AM] 5.3.1 What are the objectives? q q Based on objective, where to go next number of factors, usually a full factorial design, since the basic purpose and analysis is... selecting experimental factors and then q select the appropriate design named in section 5.3.3 that suits your objective (and follow the related links) http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri31.htm (2 of 2) [5/1/20 06 10:30:22 AM] 5.3.2 How do you select and scale the process variables? Consider adding some center points to your two-level design The term "two-level design" is something of... After factor settings are coded, center points have the value "0" Coding is described in more detail in the DOE glossary The Model or Analysis Matrix http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri32.htm (2 of 3) [5/1/20 06 10:30:22 AM] ... region where the process is easier to manage r make a process robust (note: this objective may often be accomplished with screening designs rather than with response surface designs - see Section 5.5 .6) Regression modeling r to estimate a precise model, quantifying the dependence of response variable(s) on process inputs After identifying the objective listed above that corresponds most closely to . well-behaved? http://www.itl.nist.gov/div898 /handbook/ pri/section2/pri24.htm (6 of 10) [5/1/20 06 10:30:22 AM] Interpretation of the residuals versus fitted values plots A residual distribution such as that in Figure 2 .6 showing. Jagged Responses 5.2.3. Is there a simple model? http://www.itl.nist.gov/div898 /handbook/ pri/section2/pri23.htm [5/1/20 06 10:30:20 AM] Histogram The histogram is a frequency plot obtained by placing. distribution. 5.2.4. Are the model residuals well-behaved? http://www.itl.nist.gov/div898 /handbook/ pri/section2/pri24.htm (2 of 10) [5/1/20 06 10:30:21 AM] Sample normal probability plot with overlaid dot plot Figure